I'm not sure what you are after. But this tidy up of your code may contain the magic you require:

use warnings;
use strict;

my %people;
my %families;

while(<DATA>) {
    chomp;
    my ($family, $people) = split /=/, $_, 2;
    #split into two: family and memebers
    my @members = split /,\s*/, $people, -1;         #split family lin
+e into an array
    foreach my $member (@members) {                            #foreac
+h member within array
        push @{$people{$member}}, $family;                     #push i
+nto an array
    }
    
    push @{$families{$family}}, @members;
}

my $total_families = keys %families;
my $total_people = 0;

$total_people += @{$families{$_}} for keys %families;

print "Num Families: $total_families\n";
print "Num People: $total_people\n";
print "All given Names: ", join (' ', sort keys %people), "\n";
print "All family names: ", join (' ', sort keys %families), "\n";
print "All names grouped by family:\n";
print "  ", join ', ',
    do {my $famName = $_; map {"$_ $famName"} sort @{$families{$_}};},
+ "\n"
        for sort keys %families;
        
__DATA__
Flintstone=Barney, Fred, Wilma
Jetson=Max, Tony, Wilma
Simpson=Lisa, Barney, Wilma, Homer
Alcatraz=Elijah, Max, Wilma
[download]

Prints:

Num Families: 4
Num People: 13
All given Names: Barney Elijah Fred Homer Lisa Max Tony Wilma
All family names: Alcatraz Flintstone Jetson Simpson
All names grouped by family:
  Elijah Alcatraz, Max Alcatraz, Wilma Alcatraz, 
  Barney Flintstone, Fred Flintstone, Wilma Flintstone, 
  Max Jetson, Tony Jetson, Wilma Jetson, 
  Barney Simpson, Homer Simpson, Lisa Simpson, Wilma Simpson,
[download]